
The modeling of the interaction between a poroelastic medium and a fluid in a hollow cavity is crucial for understanding, e.g., the multiphysics flow of blood and Cerebrospinal Fluid (CSF) in the brain, the supply of blood by the coronary arteries in heart perfusion, or the interaction between groundwater and rivers or lakes. In particular, the cerebral tissue's elasticity and its perfusion by blood and interstitial CSF can be described by Multi-compartment Poroelasticity (MPE), while CSF flow in the brain ventricles can be modeled by the (Navier-)Stokes equations, the overall system resulting in a coupled MPE-(Navier-)Stokes system. The aim of this paper is three-fold. First, we aim to extend a recently presented discontinuous Galerkin method on polytopal grids (PolyDG) to incorporate three-dimensional geometries and physiological interface conditions. Regarding the latter, we consider here the Beavers-Joseph-Saffman (BJS) conditions at the interface: These conditions are essential to model the friction between the fluid and the porous medium. Second, we quantitatively analyze the computational efficiency of the proposed method on a domain with small geometrical features, thus demonstrating the advantages of employing polyhedral meshes. Finally, by a comparative numerical investigation, we assess the fluid-dynamics effects of the BJS conditions and of employing either Stokes or Navier-Stokes equations to model the CSF flow. The semidiscrete numerical scheme for the coupled problem is proved to be stable and optimally convergent. Temporal discretization is obtained using Newmark's β-method for the elastodynamics equation and the θ-method for the remaining equations of the model. The theoretical error estimates are verified by numerical simulations on a test case with a manufactured solution, and a numerical investigation is carried out on a three-dimensional geometry to assess the effects of interface conditions and fluid inertia on the system.
Citation: Ivan Fumagalli. Discontinuous Galerkin method for a three-dimensional coupled fluid-poroelastic model with applications to brain fluid mechanics[J]. Mathematics in Engineering, 2025, 7(2): 130-161. doi: 10.3934/mine.2025006
[1] | Yousef Jawarneh, Humaira Yasmin, Abdul Hamid Ganie, M. Mossa Al-Sawalha, Amjid Ali . Unification of Adomian decomposition method and ZZ transformation for exploring the dynamics of fractional Kersten-Krasil'shchik coupled KdV-mKdV systems. AIMS Mathematics, 2024, 9(1): 371-390. doi: 10.3934/math.2024021 |
[2] | Ihsan Ullah, Aman Ullah, Shabir Ahmad, Hijaz Ahmad, Taher A. Nofal . A survey of KdV-CDG equations via nonsingular fractional operators. AIMS Mathematics, 2023, 8(8): 18964-18981. doi: 10.3934/math.2023966 |
[3] | Rasool Shah, Abd-Allah Hyder, Naveed Iqbal, Thongchai Botmart . Fractional view evaluation system of Schrödinger-KdV equation by a comparative analysis. AIMS Mathematics, 2022, 7(11): 19846-19864. doi: 10.3934/math.20221087 |
[4] | Aslı Alkan, Halil Anaç . The novel numerical solutions for time-fractional Fornberg-Whitham equation by using fractional natural transform decomposition method. AIMS Mathematics, 2024, 9(9): 25333-25359. doi: 10.3934/math.20241237 |
[5] | Maysaa Al-Qurashi, Saima Rashid, Fahd Jarad, Madeeha Tahir, Abdullah M. Alsharif . New computations for the two-mode version of the fractional Zakharov-Kuznetsov model in plasma fluid by means of the Shehu decomposition method. AIMS Mathematics, 2022, 7(2): 2044-2060. doi: 10.3934/math.2022117 |
[6] | Saleh Baqer, Theodoros P. Horikis, Dimitrios J. Frantzeskakis . Physical vs mathematical origin of the extended KdV and mKdV equations. AIMS Mathematics, 2025, 10(4): 9295-9309. doi: 10.3934/math.2025427 |
[7] | Azzh Saad Alshehry, Naila Amir, Naveed Iqbal, Rasool Shah, Kamsing Nonlaopon . On the solution of nonlinear fractional-order shock wave equation via analytical method. AIMS Mathematics, 2022, 7(10): 19325-19343. doi: 10.3934/math.20221061 |
[8] | M. S. Alqurashi, Saima Rashid, Bushra Kanwal, Fahd Jarad, S. K. Elagan . A novel formulation of the fuzzy hybrid transform for dealing nonlinear partial differential equations via fuzzy fractional derivative involving general order. AIMS Mathematics, 2022, 7(8): 14946-14974. doi: 10.3934/math.2022819 |
[9] | Amjad Ali, Iyad Suwan, Thabet Abdeljawad, Abdullah . Numerical simulation of time partial fractional diffusion model by Laplace transform. AIMS Mathematics, 2022, 7(2): 2878-2890. doi: 10.3934/math.2022159 |
[10] | Hilal Aydemir, Mehmet Merdan, Ümit Demir . A new approach to solving local fractional Riccati differential equations using the Adomian-Elzaki method. AIMS Mathematics, 2025, 10(4): 9122-9149. doi: 10.3934/math.2025420 |
The modeling of the interaction between a poroelastic medium and a fluid in a hollow cavity is crucial for understanding, e.g., the multiphysics flow of blood and Cerebrospinal Fluid (CSF) in the brain, the supply of blood by the coronary arteries in heart perfusion, or the interaction between groundwater and rivers or lakes. In particular, the cerebral tissue's elasticity and its perfusion by blood and interstitial CSF can be described by Multi-compartment Poroelasticity (MPE), while CSF flow in the brain ventricles can be modeled by the (Navier-)Stokes equations, the overall system resulting in a coupled MPE-(Navier-)Stokes system. The aim of this paper is three-fold. First, we aim to extend a recently presented discontinuous Galerkin method on polytopal grids (PolyDG) to incorporate three-dimensional geometries and physiological interface conditions. Regarding the latter, we consider here the Beavers-Joseph-Saffman (BJS) conditions at the interface: These conditions are essential to model the friction between the fluid and the porous medium. Second, we quantitatively analyze the computational efficiency of the proposed method on a domain with small geometrical features, thus demonstrating the advantages of employing polyhedral meshes. Finally, by a comparative numerical investigation, we assess the fluid-dynamics effects of the BJS conditions and of employing either Stokes or Navier-Stokes equations to model the CSF flow. The semidiscrete numerical scheme for the coupled problem is proved to be stable and optimally convergent. Temporal discretization is obtained using Newmark's β-method for the elastodynamics equation and the θ-method for the remaining equations of the model. The theoretical error estimates are verified by numerical simulations on a test case with a manufactured solution, and a numerical investigation is carried out on a three-dimensional geometry to assess the effects of interface conditions and fluid inertia on the system.
Due to its broad relevance and propensity to incorporate many repercussions of actual concerns, the idea of fractional calculus (FC) has garnered considerable prominence in previous decades. Classical calculus has remained a small segment of FC, despite the fact that it can demonstrate numerous critical challenges and assist us in forecasting the behaviour of intricate occurrences in impulsive integro-differential equations [1], neural networking [2], thermal energy [3], non-Newtonian fluids [4] and heat flux [5]. Despite the fact that innovators offer numerous novel concepts, several aspects should always be deduced in order to guarantee all categories of phenomenon, that will be accomplished by conquering the restrictions posed by mathematicians and scientists. This is highly pertinent when investigating of MHD electro-osmotically flow [6], epidemics [7,8,9], stability and instability of special functions [10,11,12,13], inequalities [14,15,16] as well as other disciplines. When it tends to arrive to the exploration of repercussions that assist in resolving major difficulties (such as the current global challenges), there is always room for improvements, innovation, creativeness, and extensions in analysis, and so many investigators have inferred provoking outcomes with the assistance of FC, and by incorporating efficacious methodologies with the assistance of underlying FC findings [17,18,19].
Numerical models investigation and analysis of corresponding features are often a high priority in mathematical modeling when the relevant techniques are implemented. This is certainly pertinent in epidemic research, bifurcation, thermodynamics, electrostatistics modeling, fluid flow, plasma physics, and other fields. Several approaches, including N-solitons [20,21,22,23], solitary waves [24,25,26,27], Tan-Cot function method [28], Adomian decomposition method [29], homotopy perturbation method [30], q-homotopy analysis method [31], variation iteration method [32], collocation-shooting method [33], G/G′ expansion method [34], improved tan(ϕ(τ)/2)-expansion method [35], Lie symmetry analysis method [36], wavelet method [37] have been employed and refined by researchers to achieve the analytic, semi-analytic, and numerical solution of nonlinear PDEs. The Adomian decomposition method [29,38] is one of them, and it offers an efficient approach for exact-analytical solutions across a broad and comprehensive domain of specific aspects that simulate real-world issues. This strategy transforms a basic, incredibly straightforward problem into the complicated problem under investigation, and when combined with Adomian components, it offers a tremendous mathematical instrument. Numerous aspects of the Adomian decomposition method have also received considerable focus recently.
In this analysis, we examine a nonlinear framework that explains powerful interactions between interior disturbances in the water. The Korteweg de-Vries (KdV) equations are frequently utilized to illustrate acoustic wave behaviour and its physical relevance. Due to the immense amplitude of lengthy longitudinal waves and prolonged rotating impacts, we employ a modified KdV equation. We now evaluate the following models using the isopycnic surface W(x,t), which dipicts the KdV and modified KdV equations [39], respectively:
∂W(x,t)∂t+a1W(x,t)∂W(x,t)∂x+a2W2(x,t)∂W(x,t)∂x+b1∂3W(x,t)∂x3=0, | (1.1) |
where a1 and a2 signifies the quadratic and cubic non-linear coefficients, respectively. Also, the coefficient of small-scale dispersion is denoted by b1. Here, a1 and a1 presents the proportional factors associating in the aforsaid equation, and is due to the nonlinear hydrodynamic system, and it appears classically, see[40,41].
Furthermore, we compute the exact-analytical solution of nonlinear dispersive equations K(n,n):
∂W(x,t)∂t+∂∂x(Wn)+∂3∂x3(Wn)=0,n>1. | (1.2) |
Equation (1.2) is the evolutionary model for compactons. Compactons are characterized as solitons with bounded wave lengths or solitons without exponential tails in solitary wave theory (Rosenau and Hyma, 1993). Compactons are formed by the intricate coupling of nonlinear convection ∂∂x(Wn) and nonlinear dispersion ∂3∂x3(Wn) in (1.2).
Amidst Gorge Adomian's massive boost in 1980, the Adomian decomposition method introduced a well-noted terminology. It has been intensively implemented for a diverse set of nonlinear PDEs, for instance, the Korteweg-De Vries model [42], Fisher's model [43], Zakharov–Kuznetsov equation [44] and so on. The ADM was determined to be significantly related to a variety of integral transforms, including Laplace, Swai, Mohand, Aboodh, Elzaki, and others. Humanity is continuously striving to improve performance and minimizing the method's intricacy through invention, modernity, and experimentation. In connection with this, Jafari [45] propounded a well-known integral transform which is known to generalized integral transform. The dominant feature of this transformation is that it has the ability to recapture several existing transformations, see Remark 1.
Motivated by the above propensity, we aim to establish a semi-analytical approach by mingling the Jafari transform with the Adomian decomposition method, namely the Jafari decompostion method (JDM). With the assistance of fractional derivative operators, we constructed the approximate-analytical solutions for KdV, MKdV, K(2,2) and K(3,3). The suggested methodology helps us increase flexibility in determining the initial conditions, and its novelty is that it has a straightforward solution technique. This approach is straightforward and encompasses all of JDM accomplishments, as well as encouraging several scholars to investigate a broad spectrum of applications and processes. The tool to overcome computational complexity without any constraints, perturbations, or transformations from nonlinear to linear, or partial to ordinary differential equations, is the distinctive characteristic of the proposed approach. Furthermore, it is connected to factors that are extremely useful in bringing the findings to a favourable conclusion. It also is coupled to well-posed transformation, which tries to diminish the technique's intricacy while increasing its application and dependability.
In this section, we evoke some essential concepts, notions, and definitions concerning fractional derivative operators depending on power and Mittag-Leffer as a kernel, along with the detailed consequences of the Jafari transform.
Definition 2.1. ([17]) The Caputo fractional derivative (CFD) is described as follows:
c0Dλt={1Γ(r−λ)t∫0W(r)(x)(t−x)λ+1−rdx,r−1<λ<r,drdtrW(t),λ=r. | (2.1) |
Definition 2.2. ([18]) The Atangana-Baleanu fractional derivative operator in the Caputo form (ABC) is stated as follows:
ABCη1Dλt(W(t))=A(λ)1−λt∫η1W′(t)Eλ[−λ(t−x)λ1−λ]dx, | (2.2) |
where W∈H1(a1,a2)(Sobolevspace),a1<a2,λ∈[0,1] and A(λ) signifies a normalization function as A(λ)=A(0)=A(1)=1.
Definition 2.3. ([18]) The fractional integral of the ABC-operator is described as follows:
ABCη1Iλt(W(t))=1−λA(λ)W(t)+λΓ(λ)A(λ)t∫η1W(x)(t−x)λ−1dx. | (2.3) |
Definition 2.4. ([45]) Consider an integrable mapping W(t) defined on a set P, then
P={W(t):∃M>0,κ>0,|W(t)|<Mexp(κt),ift≥0}. | (2.4) |
Definition 2.5. ([45]) Suppose the mappings ϕ(s),ψ(s):R+↦R+ such that φ(s)≠0∀s∈R+. The Jafari transform of the mapping W(t) presented by Q(s) is described as
J{W(t),s}=Q(s)=ϕ(s1)∞∫0W(t)exp(−ψ(s)t)dt. | (2.5) |
Theorem 2.6. ([45]) (Convolution property). For Jafari transform, the subsequent holds true:
J{W1∗W2}=1ϕ(s)Q1(s)∗Q2(s). | (2.6) |
Definition 2.7. The Jafari transform of the CFD operator is stated as follows:
J{c0Dλt(W(t)),s}=ψλ(s)Q(s1)−ϕ(s)λ−1∑κ=0ψλ−κ−1(s1)W(κ)(0),r−1<λ<r,ϕ,ψ>0. | (2.7) |
Remark 1. Definition 2.7 leads to the following conclusions:
1) Taking ϕ(s)=1 and ψ(s)=s, then we acquire the Laplace transform [46].
2) Taking ϕ(s)1s and ψ(s)=1s, then we acquire the α-Laplace transform [47].
3) Taking ϕ(s)=1s and ψ(s)=1s, then we acquire the Sumudu transform [48].
4) Taking ϕ(s)=1s and ψ(s)=1, then we acquire the Aboodh transform [49].
5) Taking ϕ(s)=s and ψ(s)=s2, then we acquire the Pourreza transform [50,51].
6) Taking ϕ(s)=s and ψ(s)=1s, then we acquire the Elzaki transform [52].
7) Taking ϕ(s)=u2 and ψ(s)=su2, then we acquire the Natural transform [53].
8) Taking ϕ(s)=s2 and ψ(s)=s, then we acquire the Mohand transform [54].
9) Taking ϕ(s)=1s2 and ψ(s)=1s, then we acquire the Swai transform [55].
10) Taking ϕ(s)=1 and ψ(s)=1s, then we get the Kamal transform [56].
11) Taking ϕ(s)=sα and ψ(s)=1s, then we acquire the G−transform [57,58].
Definition 2.8. ([59]) The Jafari transform of the ABC fractional derivative operator is described as:
J{ABC0Dλt(W(t)),s}(λ)=A(λ)ψλ(s)λ+(1−λ)ψλ(s)(Q(s)−ϕ(s)ψ(s)W(0)). | (2.8) |
Remark 2. Definition 2.8 leads to the following conclusions:
1) Taking ϕ(s)=1 and ψ(s)=s, then we acquire the Laplace transform of ABC fractional derivative operator [60,61].
2) Taking ϕ(s)=s and ψ(s)=1s, then we acquire the Elzaki transform of ABC fractional derivative operator [62].
3) Taking ϕ(s)=ψ(s)=1s, then we get the Sumudu transform of ABC fractional derivative operator [63].
4) Taking ϕ(s)=1 and ψ(s)=s/u2, then we get the Shehu transform of ABC fractional derivative operator [63].
Definition 2.9. ([64]) The Mittag-Leffler function for single parameter is described as
Eλ(z)=∞∑κ=0zκ1Γ(κλ+1),λ,z1∈C,ℜ(λ)≥0. | (2.9) |
Consider the generic fractional form of PDE:
DλtW(x,t)+LW(x,t)+NW(x,t)=F(x,t),t>0,0<λ≤1 | (3.1) |
with ICs
W(x,0)=G(x), | (3.2) |
where Dλt=∂λW(x,t)∂tλ symbolizes the Caputo and ABC fractional derivative of order λ∈(0,1] while L and N denotes the linear and nonlinear factors, respectively. Also, F(x,t) represents the source term.
Taking into account the Jafari transform to (3.1), and we acquire
J[DλtW(x,t)+LW(x,t)+NW(x,t)]=J[F(x,t)]. |
Firstly, applying the differentiation rule of Jafari transform with respect to CFD, then we apply the ABC fractional derivativ operator as follows:
ψλ(s)U(x,s)=ϕ(s)ℓ−1∑κ=0ψλ−1−κ(s)W(κ)(0)+J[LW(x,t)+NW(x,t)]+J[F(x,t)], | (3.3) |
and
ψλ(s)A(λ)λ+(1−λ)ψλ(s)U(x,s)=ϕ(s)ψ(s)ψλ(s)A(λ)λ+(1−λ)ψλ(s)W(0)+J[LW(x,t)+NW(x,t)]+J[F(x,t)]. | (3.4) |
The inverse Jafari transform of (3.3) and (3.4) yields
W(x,t)=J−1[ϕ(s)ℓ−1∑κ=0ψ(s)λ−κ−1W(κ)(0)+1ψλ(s)J[F(x,t)]]−J−1[1ψλ(s)J[LW(x,t)+NW(x,t)]]. | (3.5) |
and
W(x,t)=J−1[ϕ(s)ψ(s)W(0)+λ+(1−λ)ψλ(s)ψλ(s)A(λ)J[F(x,t)]]−J−1[λ+(1−λ)ψλ(s)ψλ(s)A(λ)J[LW(x,t)+NW(x,t)]]. | (3.6) |
The generalized decomposition method solution W(x,t) is represented by the following infinite series
W(x,t)=∞∑ℓ=0Wℓ(x,t). | (3.7) |
Thus, the nonlinear term N(x,t) can be evaluated by the Adomian decomposition method prescribed as
NW(x,t)=∞∑ℓ=0˜Aℓ(W0,W1,...),ℓ=0,1,..., | (3.8) |
where
˜Aℓ(W0,W1,...)=1ℓ![dℓdςℓN(∞∑ȷ=0ςȷWȷ)]ς=0,ℓ>0. |
Inserting (3.7) and (3.8) into (3.5) and (3.6), respectively, we have
∞∑ℓ=0Wℓ(x,t)=G(x)+˜G(x)−J−1[1ψλ(s)J[LW(x,t)+∞∑ℓ=0˜Aℓ]] | (3.9) |
and
∞∑ℓ=0Wℓ(x,t)=G(x)+˜G(x)−J−1[λ+(1−λ)ψλ(s)A(λ)ψλ(s)J[LW(x,t)+∞∑ℓ=0˜Aℓ]]. | (3.10) |
Consequently, the recursive technique for (3.9) and (3.10) are established as:
W0(x,t)=G(x)+˜G(x),ℓ=0,Wℓ+1(x,t)=−J−1[1ψλ(s)J[L(Wℓ(x,t))+∞∑ℓ=0˜Aℓ]],ℓ≥1,Wℓ+1(x,t)=−J−1[λ+(1−λ)ψλ(s)A(λ)ψλ(s)J[L(Wℓ(x,t))+∞∑ℓ=0˜Aℓ]],ℓ≥1. | (3.11) |
In what follows, we present the various kinds of partial differential equations with the CFD and AB-frctional derivative operators, respectively.
Example 4.1. Assume that the time-fractional KdV equation
DλtW(x,t)−6WWx(x,t)+Wxxx(x,t)=0, | (4.1) |
with IC:
W0(x,0)=−2σ2exp(σx)(1+exp(σx))2. | (4.2) |
Proof. Foremost, we provide the solution of (4.1) in two general cases.
Case Ⅰ. Firstly, we apply the Caputo fractional derivative operator coupled with the Jafari transform and Adomian decomposition method. Applying the Jafari transform to (4.1).
ψλ(s)U(x,s)−ϕ(s)m−1∑κ=0ψλ−κ−1(s)W(κ)(0)=J[6WWx(x,t)−Wxxx(x,t)]. | (4.3) |
Taking into consideration the IC given in (4.2), we have
U(x,s)=ϕ(s)ψ(s)W(x,0)+1ψλ(s)J[6WWx(x,t)−Wxxx(x,t)]. |
Employing the inverse Jafari transform, we obtain
W(x,t)=J−1[ϕ(s)ψ(s)W(x,0)+1ψλ(s)J[6WWx(x,t)−Wxxx(x,t)]]. | (4.4) |
Thanks to the JDM, we find
W0(x,t)=J−1[ϕ(s)ψ(s)W(x,0)]=−2J−1[ϕ(s)ψ(s)σ2exp(σx)(1+exp(σx))2]=−2σ2exp(σx)(1+exp(σx))2. |
Here, we surmise that the unknown function W(x,t) can be written by an infinite series of the form
W(x,t)=∞∑ℓ=0Wℓ(x,t). |
Also, the non-linearity F(W) can be decomposed by an infinite series of polynomials represented by
F(W)=WWx=∞∑ℓ=0Aℓ, |
where Wℓ(x,t) will be evaluated recurrently, and Aℓ is the so-called polynomial of W0,W1,...,Wℓ established by [65].
∞∑ℓ=0Wℓ+1(x,t)=J−1[1ψλ(s)J[6∞∑ℓ=0(A)ℓ+∞∑ℓ=0(Wxxx)ℓ]],ℓ=0,1,2,.... |
The first few Adomian polynomials are presented as follows:
Aℓ(WWx)={W0W0x,ℓ=0,W0xW1+W1xW0,ℓ=1,W2W0x+W1W1x+W0W2x,ℓ=2, | (4.5) |
For ℓ=0,1,2,3,...
W1(x,t)=J−1[1ψλ(s)J[6A0+W0xxx]]=−2σ5exp(σx)(exp(σx)−1)(1+exp(σx1))3tλΓ(λ+1),W2(x,t)=J−1[1ψλ(s)J[6A1+W1xxx]]=−2σ8exp(σx)(exp(2σx)−4exp(σx)+1)(1+exp(σx1))4t2λΓ(2λ+1),⋮. |
The approximate solution for Example 4.1 is expressed as:
W(x,t)=W0(x,t)+W1(x,t)+W2(x,t)+W3(x,t)+...,=−2σ2exp(σx)(1+exp(σx))2−2σ5exp(σx)(exp(σx)−1)(1+exp(σx1))3tλΓ(λ+1)−2σ8exp(σx)(exp(2σx)−4exp(σx)+1)(1+exp(σx1))4t2λΓ(2λ+1)+.... | (4.6) |
Case Ⅱ. Here, we surmise ABC fractional derivative operator coupled with the Jafari transform and Adomian decomposition method. Applying the Jafari transform for Example 4.1.
ψλ(s)A(λ)λ+(1−λ)ψλ(s)U(x,s)−ϕ(s)m−1∑κ=0ψλ−κ−1(s)W(κ)(0)=J[6WWx(x,t)−Wxxx(x,t)]. | (4.7) |
Taking into consideration the IC given in (4.2), we have
U(x,s)=ϕ(s)ψ(s)W(x,0)+λ+(1−λ)ψλ(s)ψλ(s)A(λ)J[6WWx(x,t)−Wxxx(x,t)]. |
Employing the inverse Jafari transform, we obtain
W(x,t)=J−1[ϕ(s)ψ(s)W(x,0)+λ+(1−λ)ψλ(s)ψλ(s)A(λ)J[6WWx(x,t)−Wxxx(x,t)]]. | (4.8) |
Thanks to the JDM, we find
W0(x,t)=J−1[ϕ(s)ψ(s)W(x,0)]=−2J−1[ψ(s1)ϕ(s)σ2exp(σx)(1+exp(σx))2]=−2σ2exp(σx)(1+exp(σx))2. |
Here, we surmise that the unknown function W(x,t) can be written by an infinite series of the form
W(x,t)=∞∑ℓ=0Wℓ(x,t). |
Also, the non-linearity F1(W) can be decomposed by an infinite series of polynomials represented by
F1(W)=WWx=∞∑ℓ=0Aℓ, |
where Wℓ(x,t) will be evaluated recurrently, and Aℓ is the so-called polynomial of W0,W1,...,Wℓ defined in (4.5). Then, we have
For ℓ=0,1,2,3,...
W1(x,t)=J−1[λ+(1−λ)ψλ(s)ψλ(s)A(λ)J[6A0+W0xxx]]=−2A(λ)σ5exp(σx)(exp(σx)−1)(1+exp(σx1))3[λtλΓ(λ+1)+(1−λ)],W2(x,t)=J−1[λ+(1−λ)ψλ(s)ψλ(s)A(λ)J[6A1+W1xxx]]=−2A2(λ)σ8exp(σx)(exp(2σx)−4exp(σx)+1)(1+exp(σx1))4[λ2t2λΓ(2λ+1)+2λ(1−λ)tλΓ(λ+1)+(1−λ)2],⋮. |
The approximate solution for Example 4.1 is expressed as:
W(x,t)=W0(x,t)+W1(x,t)+W2(x,t)+W3(x,t)+...,=−2σ2exp(σx)(1+exp(σx))2−2σ5exp(σx)(exp(σx)−1)A(λ)(1+exp(σx1))3[λtλΓ(λ+1)+(1−λ)]−2σ8exp(σx)(exp(2σx)−4exp(σx)+1)A2(λ)(1+exp(σx1))4[λ2t2λΓ(2λ+1)+2λ(1−λ)tλΓ(λ+1)+(1−λ)2]+.... | (4.9) |
For λ=1, we obtained the exact solution of Example 4.1 as
W(x,t)=−σ22sech2σ2(x−σ2t). |
Figure 1 shows the evolutionary outcomes for the explicit and approximate solutions of Example 4.1 for the particular instance λ=1. The result generated by the proposed technique is remarkably similar to the exact solution, as shown in Figure 1.
Then, by considering only the first few elements of the linear equations features are integrated, we can deduce that we have accomplished a reasonable estimation with the numerical solutions of the problem. It is obvious that by adding additional components to the decomposition series (4.6) and (4.9), the cumulative error can be diminished.
Analogously, we demonstrate the two-dimensional view of the change in fractional values of the order. We depict the response in Figure 2. It is remarkable that pairwise collisions of particle-like phenomena (including solitary waves and breathers) are fundamental mechanisms in the production of condensed soliton gas dynamics. Deep water waves, shallow groundwater waves, internally waves in a segmented sea, and fibre optics are all manifestations of these waves.
Remark 3. It is remarkable that equivalent version of the KdV equation is presented as
DλtW(x,t)+6WWx(x,t)+Wxxx(x,t)=0, |
with IC:
W0(x,0)=−2σ2exp(σx)(1+exp(σx))2. |
has the solitary wave solution, when λ=1, then
W(x,t)=σ22sech2σ2(x−σ2t). |
Example 4.2. Assume that the time-fractional modified KdV equation
DλtW(x,t)+6W2Wx(x,t)+Wxxx(x,t)=0, | (4.10) |
with IC:
W0(x,0)=2σexp(σx)1+exp(2σx). | (4.11) |
Proof. Foremost, we provide the solution of (4.10) in two general cases.
Case Ⅰ. Firstly, we apply the Caputo fractional derivative operator coupled with the Jafari transform and Adomian decomposition method.
Applying the Jafari transform to (4.10).
ψλ(s)U(x,s)−ϕ(s)m−1∑κ=0ψλ−κ−1(s)W(κ)(0)=−J[6W2Wx(x,t)+Wxxx(x,t)]. | (4.12) |
Taking into consideration the IC given in (4.11), we have
U(x,s)=ϕ(s)ψ(s)W(x,0)−1ψλ(s)J[6W2Wx(x,t)+Wxxx(x,t)]. |
Employing the inverse Jafari transform, we obtain
W(x,t)=J−1[ϕ(s)ψ(s)W(x,0)−1ψλ(s)J[6W2Wx(x,t)+Wxxx(x,t)]]. | (4.13) |
Thanks to the JDM, we find
W0(x,t)=J−1[ϕ(s)ψ(s)W(x,0)]=2J−1[ϕ(s)ψ(s)σexp(σx)1+exp(2σx)]=2σexp(σx)1+exp(2σx). |
Here, we surmise that the unknown function W(x,t) can be written by an infinite series of the form
W(x,t)=∞∑ℓ=0Wℓ(x,t). |
Also, the non-linearity F(W) can be decomposed by an infinite series of polynomials represented by
F(W)=W2Wx=∞∑ℓ=0Bℓ, |
where Wℓ(x,t) will be evaluated recurrently, and Bℓ is the so-called polynomial of W0,W1,...,Wℓ established by [65].
∞∑ℓ=0Wℓ+1(x,t)=−J−1[1ψλ(s)J[6∞∑ℓ=0(B)ℓ+∞∑ℓ=0(Wxxx)ℓ]],ℓ=0,1,2,.... |
The first few Adomian polynomials are presented as follows:
Bℓ(W2Wx)={W20W0x,ℓ=0,W0x(2W0W1)+W1xW20,ℓ=1,(2W2W0+W21)W0x+(2W0W1)W1x+W20W2x,ℓ=2, | (4.14) |
For ℓ=0,1,2,3,...
W1(x,t)=−J−1[1ψλ(s)J[6B0+W0xxx]]=−2σ4exp(σx)(1−exp(2σx))(1+exp(2σx1))2tλΓ(λ+1),W2(x,t)=−J−1[1ψλ(s)J[6B1+W1xxx]]=2σ7exp(σx)(1−6exp(2σx)+exp(4σx))(1+exp(2σx1))3t2λΓ(2λ+1),⋮. |
The approximate solution for Example 4.2 is expressed as:
W(x,t)=W0(x,t)+W1(x,t)+W2(x,t)+W3(x,t)+...,=2σexp(σx)1+exp(2σx)−2σ4exp(σx)(1−exp(2σx))(1+exp(2σx1)2tλΓ(λ+1)+2σ7exp(σx)(1−6exp(2σx)+exp(4σx))(1+exp(2σx1)3t2λΓ(2λ+1)+.... | (4.15) |
Case Ⅱ. Here, we surmise ABC fractional derivative operator coupled with the Jafari transform and Adomian decomposition method.
Applying the Jafari transform on (4.10).
ψλ(s)A(λ)λ+(1−λ)ψλ(s)U(x,s)−ϕ(s)m−1∑κ=0ψλ−κ−1(s)W(κ)(0)=−J[6W2Wx(x,t)+Wxxx(x,t)]. | (4.16) |
Taking into consideration the IC given in (4.11), we have
U(x,s)=ϕ(s)ψ(s)W(x,0)−λ+(1−λ)ψλ(s)ψλ(s)A(λ)J[6W2Wx(x,t)+Wxxx(x,t)]. |
Employing the inverse Jafari transform, we obtain
W(x,t)=J−1[ϕ(s)ψ(s)W(x,0)−λ+(1−λ)ψλ(s)ψλ(s)A(λ)J[6W2Wx(x,t)+Wxxx(x,t)]]. | (4.17) |
Thanks to the JDM, we find
W0(x,t)=J−1[ϕ(s)ψ(s)W(x,0)]=2J−1[ψ(s1)ϕ(s)σexp(σx)1+exp(2σx)]=σexp(σx)1+exp(2σx). |
Here, we surmise that the unknown function W(x,t) can be written by an infinite series of the form
W(x,t)=∞∑ℓ=0Wℓ(x,t). |
Also, the non-linearity F(W), can be decomposed by an infinite series of polynomials represented by
F1(W)=W2Wx=∞∑ℓ=0Bℓ, |
where Wℓ(x,t) will be evaluated recurrently, and Bℓ is the so-called polynomial of W0,W1,...,Wℓ defined in (4.14).
For ℓ=0,1,2,3,...
W1(x,t)=−J−1[λ+(1−λ)ψλ(s)ψλ(s)A(λ)J[6B0+W0xxx]]=−2A(λ)σ4exp(σx)(1−exp(2σx))(1+exp(2σx1)2[λtλΓ(λ+1)+(1−λ)],W2(x,t)=J−1[λ+(1−λ)ψλ(s)ψλ(s)A(λ)J[6A1+W1xxx]]=2A2(λ)σ7exp(σx)(1−6exp(2σx)+exp(4σx))(1+exp(2σx1))3×[λ2t2λΓ(2λ+1)+2λ(1−λ)tλΓ(λ+1)+(1−λ)2],⋮. |
The approximate solution for Example 4.2 is expressed as:
W(x,t)=W0(x,t)+W1(x,t)+W2(x,t)+W3(x,t)+...,=σexp(σx)1+exp(2σx)−2σ4exp(σx)(1−exp(2σx))A(λ)(1+exp(2σx1))2[λtλΓ(λ+1)+(1−λ)]+2σ7exp(σx)(1−6exp(2σx)+exp(4σx))A2(λ)(1+exp(2σx1))3×[λ2t2λΓ(2λ+1)+2λ(1−λ)tλΓ(λ+1)+(1−λ)2]+.... | (4.18) |
For λ=1, we obtained the exact solution of Example 4.2 as
W(x,t)=±σsechσ(x−σ2t). |
Figure 3 shows the evolutionary outcomes for the explicit and approximate solutions of Example 4.2 for the particular instance λ=1. The result generated by the proposed technique is remarkably similar to the exact solution, as shown in Figure 3.
Then, by considering only the first few elements of the nonlinear equations features are integrated, we can deduce that we have accomplished a reasonable estimation with the numerical solutions of the problem. It is obvious that by adding additional components to the decomposition series (4.15) and (4.18), the cumulative error can be diminished.
Analogously, we demonstrate the two-dimensional view of the change in fractional values of the order. Figure 4 depicts the response for exact CFD and AB fractional derivative operators. It is remarkable that pairwise collisions of particle-like phenomena (including solitary waves and breathers) are fundamental mechanisms in the production of condensed soliton gas dynamics. Deep water waves, shallow groundwater waves, internally waves in a segmented sea, and fibre optics are all manifestations of these waves.
Example 4.3. Assume that the time-fractional K(2,2) equation
DλtW(x,t)+(W2)x(x,t)+(W2)xxx(x,t)=0, | (4.19) |
with IC:
W0(x,0)=43σcos2(x4). | (4.20) |
Proof. Foremost, we provide the solution of (4.19) in two general cases.
Case Ⅰ. Firstly, we apply the Caputo fractional derivative operator coupled with the Jafari transform and Adomian decomposition method.
Applying the Jafari transform on (4.19).
ψλ(s)U(x,s)−ϕ(s)m−1∑κ=0ψλ−κ−1(s)W(κ)(0)=−J[(W2)x(x,t)+(W2)xxx(x,t)]. | (4.21) |
Taking into consideration the IC given in (4.20), we have
U(x,s)=ϕ(s)ψ(s)W(x,0)−1ψλ(s)J[(W2)x(x,t)+(W2)xxx(x,t)]. |
Employing the inverse Jafari transform, we obtain
W(x,t)=J−1[ϕ(s)ψ(s)W(x,0)−1ψλ(s)J[(W2)x(x,t)+(W2)xxx(x,t)]]. | (4.22) |
Thanks to the JDM, we find
W0(x,t)=J−1[ϕ(s)ψ(s)43σcos2(x4)]=2J−1[ϕ(s)ψ(s)43σcos2(x4)]=43σcos2(x4). |
Here, we surmise that the unknown function W(x,t) can be written by an infinite series of the form
W(x,t)=∞∑ℓ=0Wℓ(x,t). |
Also, the non-linearities F1(W) and F2(W) can be decomposed by an infinite series of polynomials represented by
F1(W)=(W2)x=∞∑ℓ=0Dℓ,F2(W)=(W2)xxx=∞∑ℓ=0Eℓ, |
where Wℓ(x,t) will be evaluated recurrently, and Dℓ and Eℓ are the so-called polynomial of W0,W1,...,Wℓ established by [65].
∞∑ℓ=0Wℓ+1(x,t)=−J−1[1ψλ(s)J[∞∑ℓ=0(D)ℓ+∞∑ℓ=0(E)ℓ]],ℓ=0,1,2,.... |
The first few Adomian polynomials are presented as follows:
Dℓ((W2)x)={W20x,ℓ=0,(2W0W1)x,ℓ=1,(2W2W0+W21)x,ℓ=2,Eℓ((W2)xxx)={W20xxx,ℓ=0,(2W0W1)xxx,ℓ=1,(2W2W0+W21)xxx,ℓ=2, | (4.23) |
For ℓ=0,1,2,3,...
W1(x,t)=−J−1[1ψλ(s)J[D0+E0]]=σ23sin(x2)tλΓ(λ+1),W2(x,t)=−J−1[1ψλ(s)J[D1+E1]]=−σ36sin(x2)t2λΓ(2λ+1),W3(x,t)=−J−1[1ψλ(s)J[D2+E2]]=−σ412sin(x2)t3λΓ(3λ+1)⋮. |
The approximate solution for Example 4.3 is expressed as:
W(x,t)=W0(x,t)+W1(x,t)+W2(x,t)+W3(x,t)+...,=43σcos2(x4)+σ23sin(x2)tλΓ(λ+1)−σ36sin(x2)t2λΓ(2λ+1)−σ412sin(x2)t3λΓ(3λ+1)+.... | (4.24) |
Case Ⅱ. Here, we surmise ABC fractional derivative operator coupled with the Jafari transform and Adomian decomposition method. Applying the Jafari transform for Example 4.19.
ψλ(s)A(λ)λ+(1−λ)ψλ(s)U(x,s)−ϕ(s)m−1∑κ=0ψλ−κ−1(s)W(κ)(0)=−J[(W2)x(x,t)+(W2)xxx(x,t)]. | (4.25) |
Taking into consideration the IC given in (4.20), we have
U(x,s)=ϕ(s)ψ(s)W(x,0)−λ+(1−λ)ψλ(s)ψλ(s)A(λ)J[(W2)x(x,t)+(W2)xxx(x,t)]. |
Employing the inverse Jafari transform, we obtain
W(x,t)=J−1[ϕ(s)ψ(s)W(x,0)−λ+(1−λ)ψλ(s)ψλ(s)A(λ)J[(W2)x(x,t)+(W2)xxx(x,t)]]. | (4.26) |
Thanks to the JDM, we find
W0(x,t)=J−1[ϕ(s)ψ(s)W(x,0)]=2J−1[ψ(s1)ϕ(s)43σcos2(x4)]=43σcos2(x4). |
Here, we surmise that the unknown function W(x,t) can be written by an infinite series of the form
W(x,t)=∞∑ℓ=0Wℓ(x,t). |
Also, the non-linearity Fȷ(W),ȷ=1,2 can be decomposed by an infinite series of polynomials represented by
F1(W)=(W2)x=∞∑ℓ=0Dℓ,F2(W)=(W2)xxx=∞∑ℓ=0Eℓ, |
where Wℓ(x,t) will be evaluated recurrently, and Dℓ and Eℓ are the so-called polynomial of W0,W1,...,Wℓ established defined in (4.23). Then, we have
For ℓ=0,1,2,3,...
W1(x,t)=−J−1[λ+(1−λ)ψλ(s)ψλ(s)A(λ)J[D0+E0]]=σ23A(λ)sin(x2)[λtλΓ(λ+1)+(1−λ)],W2(x,t)=J−1[λ+(1−λ)ψλ(s)ψλ(s)A(λ)J[D1+E1]]=−σ36A2(λ)sin(x2)[λ2t2λΓ(2λ+1)+2λ(1−λ)tλΓ(λ+1)+(1−λ)2],⋮. |
The approximate solution for Example 4.3 is expressed as:
W(x,t)=W0(x,t)+W1(x,t)+W2(x,t)+W3(x,t)+...,=43σcos2(x4)+σ23A(λ)sin(x2)[λtλΓ(λ+1)+(1−λ)]−σ36A2(λ)sin(x2)[λ2t2λΓ(2λ+1)+2λ(1−λ)tλΓ(λ+1)+(1−λ)2]+.... | (4.27) |
For λ=1, we obtained the closed form solution of Example 4.3 as
W(x,t)={43σcos2(x−σt4),|x−σt|≤2π,0,otherwise.. |
Figure 5 shows the evolutionary outcomes for the explicit and approximate solutions of Example 4.3 for the particular instance λ=1. The result generated by the proposed technique is remarkably similar to the exact solution, as shown in Figure 5.
Then, by considering only the first few elements of the nonlinear equations features are integrated, we can deduce that we have accomplished a reasonable estimation with the numerical solutions of the problem. It is obvious that by adding additional components to the decomposition series (4.24) and (4.27), the cumulative error can be diminished.
Analogously, we demonstrate the two-dimensional view of the change in fractional values of the order. Figure 6 depicts the response for exact CFD and AB fractional derivative operators.
For a variation of the K(2,2) equation, constrained traveling-wave solutions are achieved. We acquire hump-shaped and valley-shaped solitary-wave solutions, as well as some periodic solutions, for the focusing branch. It is worth noting that optimal focusing provides the aggregate of the frequency and amplitude of the originating waves in the engaging phase, as illustrated in reference [66].
Example 4.4. Assume that the time-fractional K(3,3) equation
DλtW(x,t)+(W3)x(x,t)+(W3)xxx(x,t)=0, | (4.28) |
with IC:
W0(x,0)=√3σ2cos(x3). | (4.29) |
Proof. Foremost, we provide the solution of (4.28) in two general cases.
Case Ⅰ. Firstly, we apply the Caputo fractional derivative operator coupled with the Jafari transform and Adomian decomposition method.
Applying the Jafari transform on (4.28).
ψλ(s)U(x,s)−ϕ(s)m−1∑κ=0ψλ−κ−1(s)W(κ)(0)=−J[(W3)x(x,t)+(W3)xxx(x,t)]. | (4.30) |
Taking into consideration the IC given in (4.29), we have
U(x,s)=ϕ(s)ψ(s)W(x,0)−1ψλ(s)J[(W3)x(x,t)+(W3)xxx(x,t)]. |
Employing the inverse Jafari transform, we obtain
W(x,t)=J−1[ϕ(s)ψ(s)W(x,0)−1ψλ(s)J[(W3)x(x,t)+(W3)xxx(x,t)]]. | (4.31) |
Thanks to the JDM, we find
W0(x,t)=J−1[ϕ(s)ψ(s)√3σ2cos(x3)]=2J−1[ϕ(s)ψ(s)√3σ2cos(x3)]=√3σ2cos(x3). |
Here, we surmise that the unknown function W(x,t) can be written by an infinite series of the form
W(x,t)=∞∑ℓ=0Wℓ(x,t). |
Also, the non-linearities F1(W) and F2(W) can be decomposed by an infinite series of polynomials represented by
F1(W)=(W3)x=∞∑ℓ=0Gℓ,F2(W)=(W3)xxx=∞∑ℓ=0Hℓ, |
where Wℓ(x,t) will be evaluated recurrently, and Gℓ and Hℓ are the so-called polynomial of W0,W1,...,Wℓ established by [65].
∞∑ℓ=0Wℓ+1(x,t)=−J−1[1ψλ(s)J[∞∑ℓ=0(G)ℓ+∞∑ℓ=0(H)ℓ]],ℓ=0,1,2,.... |
The first few Adomian polynomials are presented as follows:
Gℓ((W3)x)={W30x,ℓ=0,(3W20W1)x,ℓ=1,(3W2W20+3W21W0)x,ℓ=2,Eℓ((W3)xxx)={W20xxx,ℓ=0,(3W20W1)xxx,ℓ=1,(3W2W20+3W21W0)xxx,ℓ=2, | (4.32) |
For ℓ=0,1,2,3,...
W1(x,t)=−J−1[1ψλ(s)J[G0+H0]]=√6σ36sin(x3)tλΓ(λ+1),W2(x,t)=−J−1[1ψλ(s)J[G1+H1]]=−√6σ518sin(x3)t2λΓ(2λ+1),W3(x,t)=−J−1[1ψλ(s)J[G2+H2]]=−√6σ754sin(x3)t3λΓ(3λ+1)⋮. |
The approximate solution for Example 4.4 is expressed as:
W(x,t)=W0(x,t)+W1(x,t)+W2(x,t)+W3(x,t)+...,=√3σ2cos(x3)+√6σ36sin(x3)tλΓ(λ+1)−√6σ518sin(x3)t2λΓ(2λ+1)−√6σ754sin(x3)t3λΓ(3λ+1)+.... | (4.33) |
Case Ⅱ. Here, we surmise ABC fractional derivative operator coupled with the Jafari transform and Adomian decomposition method. Applying the Jafari transform for Example 4.28.
ψλ(s)A(λ)λ+(1−λ)ψλ(s)U(x,s)−ϕ(s)m−1∑κ=0ψλ−κ−1(s)W(κ)(0)=−J[(W3)x(x,t)+(W3)xxx(x,t)]. | (4.34) |
Taking into consideration the IC given in (4.29), we have
U(x,s)=ϕ(s)ψ(s)W(x,0)−λ+(1−λ)ψλ(s)ψλ(s)A(λ)J[(W3)x(x,t)+(W3)xxx(x,t)]. |
Employing the inverse Jafari transform, we obtain
W(x,t)=J−1[ϕ(s)ψ(s)W(x,0)−λ+(1−λ)ψλ(s)ψλ(s)A(λ)J[(W3)x(x,t)+(W3)xxx(x,t)]]. | (4.35) |
Thanks to the JDM, we find
W0(x,t)=J−1[ϕ(s)ψ(s)W(x,0)]=2J−1[ψ(s1)ϕ(s)√3σ2cos(x3)]=√3σ2cos(x3). |
Here, we surmise that the unknown function W(x,t) can be written by an infinite series of the form
W(x,t)=∞∑ℓ=0Wℓ(x,t). |
Also, the non-linearity Fȷ(W),ȷ=1,2 can be decomposed by an infinite series of polynomials represented by
F1(W)=(W3)x=∞∑ℓ=0Gℓ,F2(W)=(W3)xxx=∞∑ℓ=0Hℓ, |
where Wℓ(x,t) will be evaluated recurrently, and Dℓ and Eℓ are the so-called polynomial of W0,W1,...,Wℓ established defined in (4.32). Then, we have
For ℓ=0,1,2,3,...
W1(x,t)=−J−1[λ+(1−λ)ψλ(s)ψλ(s)A(λ)J[G0+H0]]=√6σ36A(λ)sin(x3)[λtλΓ(λ+1)+(1−λ)],W2(x,t)=J−1[λ+(1−λ)ψλ(s)ψλ(s)A(λ)J[G1+H1]]=−√6σ518A2(λ)sin(x3)[λ2t2λΓ(2λ+1)+2λ(1−λ)tλΓ(λ+1)+(1−λ)2],⋮. |
The approximate solution for Example 4.4 is expressed as:
W(x,t)=W0(x,t)+W1(x,t)+W2(x,t)+W3(x,t)+...,=√3σ2cos(x3)+√6σ36A(λ)sin(x3)[λtλΓ(λ+1)+(1−λ)]−√6σ518A2(λ)sin(x3)[λ2t2λΓ(2λ+1)+2λ(1−λ)tλΓ(λ+1)+(1−λ)2]+.... | (4.36) |
For λ=1, we obtained the closed form solution of Example 4.4 as
W(x,t)={√6σ2σcos(x−σt3),|x−σt|≤3π2,0,otherwise.. |
Figure 7 shows the evolutionary outcomes for the explicit and approximate solutions of Example 4.4 for the particular instance λ=1. The result generated by the proposed technique is remarkably similar to the exact solution, as shown in Figure 7.
Then, by considering only the first few elements of the nonlinear equations features are integrated, we can deduce that we have accomplished a reasonable estimation with the numerical solutions of the problem. It is obvious that by adding additional components to the decomposition series (4.33) and (4.36), the cumulative error can be diminished.
Analogously, we demonstrate the two-dimensional view of the change in fractional values of the order. Figure 8 depicts the response for exact CFD and AB fractional derivative operators.
For a variation of the K(3,3) equation, constrained traveling-wave solutions are achieved. We acquire hump-shaped and valley-shaped solitary-wave solutions, as well as some periodic solutions, for the focusing branch. It is worth noting that optimal focusing provides the aggregate of the frequency and amplitude of the originating waves in the engaging phase, as illustrated in reference [66].
In this paper, we conducted a novel algorithm based on the Jafari transform and Adomin decomposition method, known as the Jafari decomposition method. In the time-fractional technique, we investigated several models such as KdV, mKdV, K (2,2), and K (3,3). To comprehend their physical interpretation, we researched and examined several novel families of solutions and their simulation studies, presented in two-dimensional and three-dimensional plots. The new discoveries concerned the hyperbolic function, trigonometric function, exponential function, and constant function. These new solutions and results might be appreciated in the laser, plasma sciences and wave pattern. To summarise, the suggested method stated above was determined to solve this collection of challenges by utilizing successive fast converging approximations without any limiting requirements or manipulations that changed the physical attributes of the concerns. Also, increasing the recursive procedure leads to the closed form solution of the governing equation.
The authors declare that they have no competing interests.
[1] | M. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, et al., The FEniCS project version 1.5, Arch. Numer. Software, 3 (2015), 9–23. |
[2] |
I. Ambartsumyan, E. Khattatov, I. Yotov, P. Zunino, A Lagrange multiplier method for a Stokes-Biot fluid–poroelastic structure interaction model, Numer. Math., 140 (2018), 513–553. https://doi.org/10.1007/s00211-018-0967-1 doi: 10.1007/s00211-018-0967-1
![]() |
[3] |
D. Anderson, J. Droniou, An arbitrary-order scheme on generic meshes for miscible displacements in porous media, SIAM J. Sci. Comput., 40 (2018), B1020–B1054. https://doi.org/10.1137/17M1138807 doi: 10.1137/17M1138807
![]() |
[4] |
P. F. Antonietti, P. Houston, G. Pennesi, Fast numerical integration on polytopic meshes with applications to discontinuous Galerkin finite element methods, J. Sci. Comput., 77 (2018), 1339–1370. https://doi.org/10.1007/s10915-018-0802-y doi: 10.1007/s10915-018-0802-y
![]() |
[5] |
P. F. Antonietti, S. Bonetti, M. Botti, M. Corti, I. Fumagalli, I. Mazzieri, lymph: discontinuous poLYtopal methods for Multi-PHysics differential problems, ACM Trans. Math. Software, 51 (2025), 1–22. https://doi.org/10.1145/3716310 doi: 10.1145/3716310
![]() |
[6] | P. F. Antonietti, A. Cangiani, J. Collis, Z. Dong, E. H. Georgoulis, S. Giani, et al., Review of discontinuous Galerkin finite element methods for partial differential equations on complicated domains, In: G. Barrenechea, F. Brezzi, A. Cangiani, E. Georgoulis, Building bridges: connections and challenges in modern approaches to numerical partial differential equations, Lecture Notes in Computational Science and Engineering, Springer, 114 (2016), 281–310. https://doi.org/10.1007/978-3-319-41640-3_9 |
[7] |
P. F. Antonietti, N. Farenga, E. Manuzzi, G. Martinelli, L. Saverio, Agglomeration of polygonal grids using graph neural networks with applications to multigrid solvers, Comput. Math. Appl., 154 (2024), 45–57. https://doi.org/10.1016/j.camwa.2023.11.015 doi: 10.1016/j.camwa.2023.11.015
![]() |
[8] |
P. F. Antonietti, L. Mascotto, M. Verani, S. Zonca, Stability analysis of polytopic discontinuous Galerkin approximations of the Stokes problem with applications to fluid–structure interaction problems, J. Sci. Comput., 90 (2022), 23. https://doi.org/10.1007/s10915-021-01695-6 doi: 10.1007/s10915-021-01695-6
![]() |
[9] |
P. Antonietti, I. Mazzieri, High-order discontinuous Galerkin methods for the elastodynamics equation on polygonal and polyhedral meshes, Comput. Methods Appl. Mech. Eng., 342 (2018), 414–437. https://doi.org/10.1016/j.cma.2018.08.012 doi: 10.1016/j.cma.2018.08.012
![]() |
[10] |
A. Bacyinski, M. Xu, W. Wang, J. Hu, The paravascular pathway for brain waste clearance: current understanding, significance and controversy, Front. Neuroanat., 11 (2017), 101. https://doi.org/10.3389/fnana.2017.00101 doi: 10.3389/fnana.2017.00101
![]() |
[11] |
S. Badia, A. Quaini, A. Quarteroni, Coupling Biot and Navier-Stokes equations for modelling fluid–poroelastic media interaction, J. Comput. Phys., 228 (2009), 7986–8014. https://doi.org/10.1016/j.jcp.2009.07.019 doi: 10.1016/j.jcp.2009.07.019
![]() |
[12] | O. Balédent, M. Henry-Feugeas, I. Idy-Peretti, Cerebrospinal fluid dynamics and relation with blood flow: a magnetic resonance study with semiautomated cerebrospinal fluid segmentation, Invest. Radiol., 36 (2001), 368–377. |
[13] |
N. A. Barnafi Wittwer, S. D. Gregorio, L. Dede', P. Zunino, C. Vergara, A. Quarteroni, A multiscale poromechanics model integrating myocardial perfusion and the epicardial coronary vessels, SIAM J. Appl. Math., 82 (2022), 1167–1193. https://doi.org/10.1137/21M1424482 doi: 10.1137/21M1424482
![]() |
[14] |
G. S. Beavers, D. D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), 197–207. https://doi.org/10.1017/S0022112067001375 doi: 10.1017/S0022112067001375
![]() |
[15] |
L. Bociu, S. Canic, B. Muha, J. T. Webster, Multilayered poroelasticity interacting with Stokes flow, SIAM J. Math. Anal., 53 (2021), 6243–6279. https://doi.org/10.1137/20M1382520 doi: 10.1137/20M1382520
![]() |
[16] |
D. Boffi, M. Botti, D. A. Di Pietro, A nonconforming high-order method for the Biot problem on general meshes, SIAM J. Sci. Comput., 38 (2016), A1508–A1537. https://doi.org/10.1137/15M1025505 doi: 10.1137/15M1025505
![]() |
[17] |
W. M. Boon, M. Kuchta, K. A. Mardal, R. Ruiz-Baier, Robust preconditioners for perturbed saddle-point problems and conservative discretizations of Biot's equations utilizing total pressure, SIAM J. Sci. Comput., 43 (2021), B961–B983. https://doi.org/10.1137/20M1379708 doi: 10.1137/20M1379708
![]() |
[18] |
J. W. Both, N. A. Barnafi, F. A. Radu, P. Zunino, A. Quarteroni, Iterative splitting schemes for a soft material poromechanics model, Comput. Methods Appl. Mech. Eng., 388 (2022), 114183. https://doi.org/10.1016/j.cma.2021.114183 doi: 10.1016/j.cma.2021.114183
![]() |
[19] | L. Botti, M. Botti, D. A. Di Pietro, A hybrid high-order method for multiple-network poroelasticity, In: D. A. Di Pietro, L. Formaggia, R. Masson, Polyhedral methods in geosciences, Springer, 27 (2021), 227–258. https://doi.org/10.1007/978-3-030-69363-3_6 |
[20] |
G. S. Brennan, T. B. Thompson, H. Oliveri, M. E. Rognes, A. Goriely, The role of clearance in neurodegenerative diseases, SIAM J. Appl. Math., 84 (2024), S172–S198. https://doi.org/10.1137/22M1487801 doi: 10.1137/22M1487801
![]() |
[21] |
M. Bucelli, A. Zingaro, P. C. Africa, I. Fumagalli, L. Dede', A. Quarteroni, A mathematical model that integrates cardiac electrophysiology, mechanics, and fluid dynamics: application to the human left heart, Int. J. Numer. Methods Biomed. Eng., 39 (2023), e3678. https://doi.org/10.1002/cnm.3678 doi: 10.1002/cnm.3678
![]() |
[22] |
S. Budday, G. Sommer, J. Haybaeck, P. Steinmann, G. A. Holzapfel, E. Kuhl, Rheological characterization of human brain tissue, Acta Biomater., 60 (2017), 315–329. https://doi.org/10.1016/j.actbio.2017.06.024 doi: 10.1016/j.actbio.2017.06.024
![]() |
[23] |
M. Bukač, I. Yotov, P. Zunino, An operator splitting approach for the interaction between a fluid and a multilayered poroelastic structure, Numer. Meth. Part. D. E., 31 (2015), 1054–1100. https://doi.org/10.1002/num.21936 doi: 10.1002/num.21936
![]() |
[24] |
A. Cangiani, E. H. Georgoulis, P. Houston, hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes, Math. Mod. Meth. Appl. Sci., 24 (2014), 2009–2041. https://doi.org/10.1142/S0218202514500146 doi: 10.1142/S0218202514500146
![]() |
[25] |
M. Causemann, V. Vinje, M. E. Rognes, Human intracranial pulsatility during the cardiac cycle: a computational modelling framework, Fluids Barriers CNS, 19 (2022), 84. https://doi.org/10.1186/s12987-022-00376-2 doi: 10.1186/s12987-022-00376-2
![]() |
[26] |
X. Chen, F. Ti, M. Li, S. Liu, T. J. Lu, Theory of fluid saturated porous media with surface effects, J. Mech. Phys. Solids, 151 (2021), 104392. https://doi.org/10.1016/j.jmps.2021.104392 doi: 10.1016/j.jmps.2021.104392
![]() |
[27] |
S. W. Cheung, E. Chung, H. H. Kim, Y. Qian, Staggered discontinuous Galerkin methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 302 (2015), 251–266. https://doi.org/10.1016/j.jcp.2015.08.024 doi: 10.1016/j.jcp.2015.08.024
![]() |
[28] |
D. Chou, J. C. Vardakis, L. Guo, B. J. Tully, Y. Ventikos, A fully dynamic multi-compartmental poroelastic system: application to aqueductal stenosis, J. Biomech., 49 (2016), 2306–2312. https://doi.org/10.1016/j.jbiomech.2015.11.025 doi: 10.1016/j.jbiomech.2015.11.025
![]() |
[29] |
M. Corti, P. F. Antonietti, L. Dede', A. M. Quarteroni, Numerical modelling of the brain poromechanics by high-order discontinuous Galerkin methods, Math. Mod. Meth. Appl. Sci., 33 (2023), 1577–1609. https://doi.org/10.1142/S0218202523500367 doi: 10.1142/S0218202523500367
![]() |
[30] |
F. Dassi, D. Mora, C. Reales, I. Velásquez, Mixed variational formulations of virtual elements for the polyharmonic operator (−Δ)n, Comput. Math. Appl., 158 (2024), 150–166. https://doi.org/10.1016/j.camwa.2024.01.013 doi: 10.1016/j.camwa.2024.01.013
![]() |
[31] |
S. Deparis, G. Grandperrin, A. Quarteroni, Parallel preconditioners for the unsteady Navier-Stokes equations and applications to hemodynamics simulations, Comput. Fluids, 92 (2014), 253–273. https://doi.org/10.1016/j.compfluid.2013.10.034 doi: 10.1016/j.compfluid.2013.10.034
![]() |
[32] |
A. Dereims, S. Drapier, J. M. Bergheau, P. De Luca, 3D robust iterative coupling of Stokes, Darcy and solid mechanics for low permeability media undergoing finite strains, Finite Elem. Anal. Des., 94 (2015), 1–15. https://doi.org/10.1016/j.finel.2014.09.003 doi: 10.1016/j.finel.2014.09.003
![]() |
[33] |
S. Di Gregorio, M. Fedele, G. Pontone, A. F. Corno, P. Zunino, C. Vergara, et al., A computational model applied to myocardial perfusion in the human heart: from large coronaries to microvasculature, J. Comput. Phys., 424 (2021), 109836. https://doi.org/10.1016/j.jcp.2020.109836 doi: 10.1016/j.jcp.2020.109836
![]() |
[34] |
M. Discacciati, E. Miglio, A. Quarteroni, Mathematical and numerical models for coupling surface and groundwater flows, Appl. Numer. Math., 43 (2002), 57–74. https://doi.org/10.1016/S0168-9274(02)00125-3 doi: 10.1016/S0168-9274(02)00125-3
![]() |
[35] |
I. N. Drøsdal, K. A. Mardal, K. Støverud, V. Haughton, Effect of the central canal in the spinal cord on fluid movement within the cord, Neuroradiology J., 26 (2013), 585–590. https://doi.org/10.1177/197140091302600513 doi: 10.1177/197140091302600513
![]() |
[36] |
E. Eliseussen, M. E. Rognes, T. B. Thompson, A posteriori error estimation and adaptivity for multiple-network poroelasticity, ESAIM: M2AN, 57 (2023), 1921–1952. https://doi.org/10.1051/m2an/2023033 doi: 10.1051/m2an/2023033
![]() |
[37] |
M. Esmaily Moghadam, Y. Bazilevs, T. Y. Hsia, I. E. Vignon-Clementel, A. L. Marsden, Modeling of Congenital Hearts Alliance (MOCHA), A comparison of outlet boundary treatments for prevention of backflow divergence with relevance to blood flow simulations, Comput. Mech., 48 (2011), 277–291. https://doi.org/10.1007/s00466-011-0599-0 doi: 10.1007/s00466-011-0599-0
![]() |
[38] |
M. Feder, A. Cangiani, L. Heltai, R3MG: R-tree based agglomeration of polytopal grids with applications to multilevel methods, J. Comput. Phys., 526 (2025), 113773. https://doi.org/10.1016/j.jcp.2025.113773 doi: 10.1016/j.jcp.2025.113773
![]() |
[39] |
M. Ferronato, A. Franceschini, C. Janna, N. Castelletto, H. A. Tchelepi, A general preconditioning framework for coupled multiphysics problems with application to contact-and poro-mechanics, J. Comput. Phys., 398 (2019), 108887. https://doi.org/10.1016/j.jcp.2019.108887 doi: 10.1016/j.jcp.2019.108887
![]() |
[40] |
I. Fumagalli, M. Corti, N. Parolini, P. F. Antonietti, Polytopal discontinuous Galerkin discretization of brain multiphysics flow dynamics, J. Comput. Phys., 513 (2024), 113115. https://doi.org/10.1016/j.jcp.2024.113115 doi: 10.1016/j.jcp.2024.113115
![]() |
[41] |
I. Fumagalli, M. Fedele, C. Vergara, L. Dede', S. Ippolito, F. Nicolò, et al., An image-based computational hemodynamics study of the systolic anterior motion of the mitral valve, Comput. Biol. Med., 123 (2020), 103922. https://doi.org/10.1016/j.compbiomed.2020.103922 doi: 10.1016/j.compbiomed.2020.103922
![]() |
[42] |
C. Geuzaine, J. F. Remacle, Gmsh: a 3-D finite element mesh generator with built-in pre-and post-processing facilities, Int. J. Numer. Meth. Eng., 79 (2009), 1309–1331. https://doi.org/10.1002/nme.2579 doi: 10.1002/nme.2579
![]() |
[43] |
L. Guo, J. C. Vardakis, T. Lassila, M. Mitolo, N. Ravikumar, D. Chou, et al., Subject-specific multi-poroelastic model for exploring the risk factors associated with the early stages of alzheimer's disease, Interface Focus, 8 (2018), 20170019. https://doi.org/10.1098/rsfs.2017.0019 doi: 10.1098/rsfs.2017.0019
![]() |
[44] | L. M. Hablitz, M. Nedergaard, The glymphatic system, Curr. Biol., 31 (2021), R1371–R1375. |
[45] |
M. Hirschhorn, V. Tchantchaleishvili, R. Stevens, J. Rossano, A. Throckmorton, Fluid–structure interaction modeling in cardiovascular medicine – a systematic review 2017–2019, Med. Eng. Phys., 78 (2020), 1–13. https://doi.org/10.1016/j.medengphy.2020.01.008 doi: 10.1016/j.medengphy.2020.01.008
![]() |
[46] |
K. E. Holter, B. Kehlet, A. Devor, T. J. Sejnowski, A. M. Dale, S. W. Omholt, et al., Interstitial solute transport in 3D reconstructed neuropil occurs by diffusion rather than bulk flow, Proceedings of the National Academy of Sciences, 114 (2017), 9894–9899. https://doi.org/10.1073/pnas.1706942114 doi: 10.1073/pnas.1706942114
![]() |
[47] |
G. Karypis, V. Kumar, A fast and high quality multilevel scheme for partitioning irregular graphs, SIAM J. Sci. Comput., 20 (1998), 359–392. https://doi.org/10.1137/S1064827595287997 doi: 10.1137/S1064827595287997
![]() |
[48] |
H. H. Kim, E. T. Chung, C. S. Lee, A staggered discontinuous Galerkin method for the Stokes system, SIAM J. Numer. Anal., 51 (2013), 3327–3350. https://doi.org/10.1137/120896037 doi: 10.1137/120896037
![]() |
[49] |
W. J. Layton, F. Schieweck, I. Yotov, Coupling fluid flow with porous media flow, SIAM J. Numer. Anal., 40 (2002), 2195–2218. https://doi.org/10.1137/S0036142901392766 doi: 10.1137/S0036142901392766
![]() |
[50] |
J. J. Lee, E. Piersanti, K. A. Mardal, M. E. Rognes, A mixed finite element method for nearly incompressible multiple-network poroelasticity, SIAM J. Sci. Comput., 41 (2019), A722–A747. https://doi.org/10.1137/18M1182395 doi: 10.1137/18M1182395
![]() |
[51] |
J. Li, B. Riviere, High order discontinuous Galerkin method for simulating miscible flooding in porous media, Comput. Geosci., 19 (2015), 1251–1268. https://doi.org/10.1007/s10596-015-9541-4 doi: 10.1007/s10596-015-9541-4
![]() |
[52] |
K. Lipnikov, D. Vassilev, I. Yotov, Discontinuous Galerkin and mimetic finite difference methods for coupled Stokes-Darcy flows on polygonal and polyhedral grids, Numer. Math., 126 (2014), 321–360. https://doi.org/10.1007/s00211-013-0563-3 doi: 10.1007/s00211-013-0563-3
![]() |
[53] | A. Logg, K. A. Mardal, G. Wells, Automated solution of differential equations by the finite element method, The FEniCS book, Vol. 84, Springer Science & Business Media, 2012. https://doi.org/10.1007/978-3-642-23099-8 |
[54] |
K. A. Mardal, M. E. Rognes, T. B. Thompson, Accurate discretization of poroelasticity without Darcy stability: Stokes-Biot stability revisited, BIT Numer. Math., 61 (2021), 941–976. https://doi.org/10.1007/s10543-021-00849-0 doi: 10.1007/s10543-021-00849-0
![]() |
[55] | K. A. Mardal, M. E. Rognes, T. B. Thompson, L. M. Valnes, Mathematical modeling of the human brain, From Magnetic Resonance Images to Finite Element Simulation, Vol. 10, Springer, 2022. https://doi.org/10.1007/978-3-030-95136-8 |
[56] |
K. A. Mardal, R. Winther, Uniform preconditioners for the time dependent Stokes problem, Numer. Math., 98 (2004), 305–327. https://doi.org/10.1007/s00211-004-0529-6 doi: 10.1007/s00211-004-0529-6
![]() |
[57] |
C. Michler, A. N. Cookson, R. Chabiniok, E. Hyde, J. Lee, M. Sinclair, et al., A computationally efficient framework for the simulation of cardiac perfusion using a multi-compartment Darcy porous-media flow model, Int. J. Numer. Meth. Biomed. Eng., 29 (2013), 217–232. https://doi.org/10.1002/cnm.2520 doi: 10.1002/cnm.2520
![]() |
[58] |
A. S. Mijailovic, S. Galarza, S. Raayai-Ardakani, N. P. Birch, J. D. Schiffman, A. J. Crosby, et al., Localized characterization of brain tissue mechanical properties by needle induced cavitation rheology and volume controlled cavity expansion, J. Mech. Behav. Biomed. Mater., 114 (2021), 104168. https://doi.org/10.1016/j.jmbbm.2020.104168 doi: 10.1016/j.jmbbm.2020.104168
![]() |
[59] | A. Quarteroni, L. Dede', A. Manzoni, C. Vergara, Mathematical modelling of the human cardiovascular system: data, numerical approximation, clinical applications, Vol. 33, Cambridge University Press, 2019. https://doi.org/10.1017/9781108616096 |
[60] |
S. Saeidi, M. P. Kainz, M. Dalbosco, M. Terzano, G. A. Holzapfel, Histology-informed multiscale modeling of human brain white matter, Sci. Rep., 13 (2023), 19641. https://doi.org/10.1038/s41598-023-46600-3 doi: 10.1038/s41598-023-46600-3
![]() |
[61] |
P. G. Saffman, On the boundary condition at the surface of a porous medium, Stud. Appl. Math., 50 (1971), 93–101. https://doi.org/10.1002/sapm197150293 doi: 10.1002/sapm197150293
![]() |
[62] | E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, 1970. |
[63] |
S. Sun, M. F. Wheeler, Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media, SIAM J. Numer. Anal., 43 (2005), 195–219. https://doi.org/10.1137/S003614290241708X doi: 10.1137/S003614290241708X
![]() |
[64] |
B. Sweetman, M. Xenos, L. Zitella, A. A. Linninger, Three-dimensional computational prediction of cerebrospinal fluid flow in the human brain, Comput. Biol. Med., 41 (2011), 67–75. https://doi.org/10.1016/j.compbiomed.2010.12.001 doi: 10.1016/j.compbiomed.2010.12.001
![]() |
[65] | R. Temam, Navier-Stokes equations: theory and numerical analysis, Vol. 343, American Mathematical Society, 2001. |
[66] |
J. Tithof, K. A. Boster, P. A. Bork, M. Nedergaard, J. H. Thomas, D. H. Kelley, A network model of glymphatic flow under different experimentally-motivated parametric scenarios, iScience, 25 (2022), 104258. https://doi.org/10.1016/j.isci.2022.104258 doi: 10.1016/j.isci.2022.104258
![]() |
[67] |
E. J. Ulbrich, C. Schraner, C. Boesch, J. Hodler, A. Busato, S. E. Anderson, et al., Normative MR cervical spinal canal dimensions, Radiology, 271 (2014), 172–182. https://doi.org/10.1148/radiol.13120370 doi: 10.1148/radiol.13120370
![]() |
[68] |
J. Wen, Y. He, A strongly conservative finite element method for the coupled Stokes-Biot model, Comput. Math. Appl., 80 (2020), 1421–1442. https://doi.org/10.1016/j.camwa.2020.07.001 doi: 10.1016/j.camwa.2020.07.001
![]() |
[69] |
L. Zhao, E. T. Chung, E. J. Park, G. Zhou, Staggered DG method for coupling of the Stokes and Darcy–Forchheimer problems, SIAM J. Numer. Anal., 59 (2021), 1–31. https://doi.org/10.1137/19M1268525 doi: 10.1137/19M1268525
![]() |
[70] | S. Zonca, P. F. Antonietti, C. Vergara, A polygonal discontinuous Galerkin formulation for contact mechanics in fluid-structure interaction problems, Commun. Comput. Phys., 30 (2021), 1–33. |
[71] |
S. Zonca, C. Vergara, L. Formaggia, An unfitted formulation for the interaction of an incompressible fluid with a thick structure via an XFEM/DG approach, SIAM J. Sci. Comput., 40 (2018), B59–B84. https://doi.org/10.1137/16M1097602 doi: 10.1137/16M1097602
![]() |
1. | Saima Rashid, Muhammad Kashif Iqbal, Ahmed M. Alshehri, Rehana Ashraf, Fahd Jarad, A comprehensive analysis of the stochastic fractal–fractional tuberculosis model via Mittag-Leffler kernel and white noise, 2022, 39, 22113797, 105764, 10.1016/j.rinp.2022.105764 | |
2. | Pooyan Alinaghi Hosseinabadi, Ali Soltani Sharif Abadi, Hemanshu Pota, Sundarapandian Vaidyanathan, Saad Mekhilef, Kamal Shah, Adaptive Finite-Time Sliding Mode Backstepping Controller for Double-Integrator Systems with Mismatched Uncertainties and External Disturbances, 2022, 2022, 1607-887X, 1, 10.1155/2022/3758220 | |
3. | Saima Rashid, Fahd Jarad, Stochastic dynamics of the fractal-fractional Ebola epidemic model combining a fear and environmental spreading mechanism, 2023, 8, 2473-6988, 3634, 10.3934/math.2023183 | |
4. | Timilehin Kingsley Akinfe, Adedapo Chris Loyinmi, An improved differential transform scheme implementation on the generalized Allen–Cahn equation governing oil pollution dynamics in oceanography, 2022, 6, 26668181, 100416, 10.1016/j.padiff.2022.100416 | |
5. | Maysaa Al-Qurashi, Sobia Sultana, Shazia Karim, Saima Rashid, Fahd Jarad, Mohammed Shaaf Alharthi, Identification of numerical solutions of a fractal-fractional divorce epidemic model of nonlinear systems via anti-divorce counseling, 2022, 8, 2473-6988, 5233, 10.3934/math.2023263 | |
6. | Jarunee Soontharanon, Muhammad Aamir Ali, Hüseyin Budak, Pinar Kösem, Kamsing Nonlaopon, Thanin Sitthiwirattham, Behrouz Parsa Moghaddam, Some New Generalized Fractional Newton’s Type Inequalities for Convex Functions, 2022, 2022, 2314-8888, 1, 10.1155/2022/6261970 | |
7. | Saima Rashid, Saad Ihsan Butt, Zakia Hammouch, Ebenezer Bonyah, Alexander Meskhi, An Efficient Method for Solving Fractional Black-Scholes Model with Index and Exponential Decay Kernels, 2022, 2022, 2314-8888, 1, 10.1155/2022/2613133 | |
8. | Saima Rashid, Fahd Jarad, Hajid Alsubaie, Ayman A. Aly, Ahmed Alotaibi, A novel numerical dynamics of fractional derivatives involving singular and nonsingular kernels: designing a stochastic cholera epidemic model, 2023, 8, 2473-6988, 3484, 10.3934/math.2023178 | |
9. | Khadija Aayadi, Khalid Akhlil, Sultana Ben Aadi, Hicham Mahdioui, Weak solutions to the time-fractional g-Bénard equations, 2022, 2022, 1687-2770, 10.1186/s13661-022-01649-3 | |
10. | Saima Rashid, Bushra Kanwal, Muhammad Attique, Ebenezer Bonyah, Ibrahim Mahariq, An Efficient Technique for Time-Fractional Water Dynamics Arising in Physical Systems Pertaining to Generalized Fractional Derivative Operators, 2022, 2022, 1563-5147, 1, 10.1155/2022/7852507 | |
11. | Haresh P. Jani, Twinkle R. Singh, Study of concentration arising in longitudinal dispersion phenomenon by Aboodh transform homotopy perturbation method, 2022, 8, 2349-5103, 10.1007/s40819-022-01363-9 | |
12. | Maysaa Al Qurashi, Saima Rashid, Fahd Jarad, A computational study of a stochastic fractal-fractional hepatitis B virus infection incorporating delayed immune reactions via the exponential decay, 2022, 19, 1551-0018, 12950, 10.3934/mbe.2022605 | |
13. | Kingsley Timilehin Akinfe, A reliable analytic technique for the modified prototypical Kelvin–Voigt viscoelastic fluid model by means of the hyperbolic tangent function, 2023, 7, 26668181, 100523, 10.1016/j.padiff.2023.100523 | |
14. | Rachid Belgacem, Ahmed Bokhari, Dumitru Baleanu, Salih Djilali, New generalized integral transform via Dzherbashian--Nersesian fractional operator, 2024, 14, 2146-5703, 90, 10.11121/ijocta.1449 | |
15. | Emmanuel Kengne, Conformable derivative in a nonlinear dispersive electrical transmission network, 2024, 112, 0924-090X, 2139, 10.1007/s11071-023-09121-2 | |
16. | Nan Jiang, Research on stability and control strategies of fractional-order differential equations in nonlinear dynamic systems, 2025, 1472-7978, 10.1177/14727978251346078 |